Paper: arXiv 2505.08623

Authors: Antoine Jacquier, Adriano Oliveri Orioles, Zan Zuric

Abstract

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail – the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical extension of the rough Bergomi model, providing semi-closed and asymptotic formulae. It includes numerical results and calibration, demonstrating a good balance of theoretical depth and empirical application. The novelty lies in the generalized grey Brownian motion and its application to joint SPX/VIX options calibration.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Rough Bergomi for Joint SPX/VIX Calibration] --> B{Methodology: Introduce Generalized Grey Brownian Motion};
    B --> C[Model Development: Generalized Grey Bergomi (GGB) Model];
    C --> D{Computational Processes: Derivation of Semi-closed & Asymptotic Formulae};
    D --> E[Data/Inputs: SPX & VIX Options Data];
    E --> F[Validation & Calibration: Numerical Analysis & Calibration Results];
    F --> G[Key Outcomes: Tractable Model, Improved Calibration, Stochastic Volatility of Volatility Features];

Cited References (84)

  1. On VIX F utures in the rough Bergomi model (2018)
  2. Hybrid scheme for B rownian semistationary processes (2017)
  3. Handbook of Mathematical Functions (1965)
  4. The eSSVI volatility surface (2016)
  5. An explicit formula for the VIX in eSSVI (2016)
  6. Arbitrage-free SVI volatility surfaces (2013) — link
  7. DOUBLE THE FUN WITH CBOE’s VVIX Index (2014)
  8. Markovian approximations of stochastic V olterra equations with the fractional kernel (2023)
and 76 more
  1. Markovian Approximation of the Rough B ergomi Model for M onte C arlo Option Pricing (2021)
  2. Approximation of Some G aussian Processes (2000)
  3. Is Gauss Quadrature Better than Clenshaw-Curtis? (2008) — link
  4. A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in P hysics (2009)
  5. Fractional Brownian motion and long-range dependence (2002)
  6. Linear Estimation of Self-Similar Processes via L amperti’s Transformation (2000)
  7. Smile Dynamics IV (2009)
  8. Yet another analysis of the SP 500 at-the-money skew: crossover of different power-law behaviours (2024)
  9. Computing the SSR — → 2406.16131
  10. On the integrability of the supremum of stochastic volatility models and other martingales (2026) — link
  11. The quadratic rough H eston model and the joint S&P 500/VIX smile calibration problem (2020)
  12. Deep learning volatility (2021)
  13. Markov Processes for Stochastic Modeling, Second Edition (2013)
  14. On the martingale property in the rough B ergomi model (2019)
  15. Correlations and bounds for stochastic volatility models (2007)
  16. Large deviations for grey G aussian processes (2022) — link
  17. A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics (2009) — link
  18. Characterizations and simulations of a class of stochastic processes to model anomalous diffusion (2008)
  19. Integral representation of generalized grey B rownian motion (2020)
  20. Option valuation using the fast Fourier transform (1999) — link
  21. Pricing under rough volatility (2015) — link
  22. Singularity of generalized grey B rownian motion and time-changed B rownian motion (2020)
  23. Singularity of generalized grey B rownian motions with different parameters (2018)
  24. Some remarks on estimate of M ittag- L effler function (2019) — link
  25. The moment formula for implied volatility at extreme strikes (2004) — link
  26. Volatility is rough (2018) — link
  27. A singular integral equation with a generalized Mittag-Leffler function in the kernel (1971)
  28. The M - W right function in time-fractional diffusion processes: a tutorial survey (2010) — link
  29. H eston stochastic vol-of-vol model for joint calibration of VIX and S & P 500 options (2018) — link
  30. Malliavin Calculus in Finance: Theory and Practice (2021)
  31. Rough multifactor volatility for SPX and VIX options (2025)
  32. The Malliavin Calculus and Related Topics (2006)
  33. Models of anomalous diffusion: the subdiffusive case (2005)
  34. Arbitrage with fractional B rownian motion (1997)
  35. Are Classes of Deterministic Integrands for Fractional B rownian Motion on an Interval Complete? (2001) — link
  36. Conditional Distributions of Processes Related to Fractional B rownian Motion (2013) — link
  37. Stochastic volatility of volatility in continuous time (2009)
  38. Stochastic Volatility of Volatility and Variance Risk Premia (2013) — link
  39. Regime-switching stochastic volatility model: estimation and calibration to VIX options (2017) — link
  40. Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets (2022) — link
  41. Volatility is (mostly) path-dependent (2023)
  42. The Sum of log- N ormal Probability Distributions in Scatter Transmission Systems (1960) — link
  43. A Suggestion for Using Powerful and Informative Tests of N ormality (1990) — link
  44. Distribution of the kurtosis statistic $b_2$ for N ormal samples (1983) — link
  45. An Omnibus Test of N ormality for Moderate and Large Size Samples (1971) — link
  46. Tests for Departure from N ormality: empirical Results for the Distributions of $b_2$ and $\sqrt b_1 $ (1973) — link
  47. Cameron–Martin Type Theorem for a Class of non-Gaussian Measures (2023) — link
  48. Multifactor approximation of rough volatility models (2019)
  49. Volatility is rough (2018)
  50. Does the Term-Structure of Equity At-the-Money Skew Really Follow a Power Law? (2022)
  51. Volatility is (mostly) path-dependent (2023)
  52. Volatility models in practice: Rough, Path-dependent or M arkovian? — → 2401.03345
  53. The quintic O rnstein- U hlenbeck volatility model that jointly calibrates SPX & VIX smiles (July 2023)
  54. Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets (2022)
  55. Smile Dynamics II (2005)
  56. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  57. Long-term storage capacity of reservoirs (1951)
  58. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum (1940)
  59. Fractional B rownian motions, fractional noises and applications (1968)
  60. Long memory continuous time models (1996)
  61. Hedging options in market models modulated by the fractional B rownian motion (2001)
  62. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007)
  63. Asymptotic analysis for stochastic volatility: martingale expansion (2011)
  64. The characteristic function of rough H eston models (2019)
  65. Perfect hedging in rough H eston models (2018)
  66. Volatility options in rough volatility models (2020)
  67. Stochastic volatility of volatility and variance risk premia (2012)
  68. Volatility-of-volatility risk (2019)
  69. State spaces of multifactor approximations of nonnegative V olterra processes (2024)
  70. The characteristic function of G aussian stochastic volatility models: an analytic expression (2022)
  71. Rough PDEs for local stochastic volatility models — → 2307.09216
  72. Smile Dynamics and Rough Volatility (2024)
  73. Estimation and Simulation for the M-Wright Function (2012) — link
  74. Fractional P oisson L aw (2000) — link
  75. MWright: Mainardi-Wright Family of Distributions (2019-08-08) — link
  76. Rough V olatility (2023)

3 of 84 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.